The two fundamental operations: Multiplication and Addition.

Not until I got to graduate school did I learn the concept of a Ring. It seems a bit abstract until I realized how fundamental it is-- more on that in a second.

A Ring has two fundamental operations, multiplication and addition. These are two separate fundamental operations on numbers, and mathematicians don't fully understand their relationship (See the links in this post on the abc conjecture). However, we are taught in school that multiplication is repeated addition: 5 times 3 is just 5 added 3 times. While this is true, it's only true in the special case of multiplying whole numbers. 16.4 times pi is not calculated by repeated addition.

I'm writing this now because I recently read this excellent blog post on exactly this point. I then read a very interesting response by a math teacher, who of course himself learned multiplication as repeated addition. Finally there is a follow-up written by the first blog post author.

One of my favorite quotes from Devlin, the original author, is "No wonder so many people end up thinking mathematics is just a bunch of arbitrary, illogical rules that cannot be figured out but simply have to be learned - only for them to have the rug pulled from under them when the rule they just learned is replaced by some other (seemingly) arbitrary, illogical rule."

I think that we should definitely avoid this approach to teaching math. Of course, perhaps some of my peers and I would have loved to learn abstract algebra in grade school-- but presumably, not everyone wants to. Up until high school, I do think that we should focus on teaching what is most practical. But it would be nice if in topics like this, we could teach the more fundamental way, and so both practically-minded students and academic-minded students would do better as they continue to learn more math.

For those who are interested, a Ring is an abstract concept of a set of objects with two operations: addition and multiplication. The set of objects must be "closed under those operations," meaning that if you add or multiply two objects, the result is also in the set. So the integers are a ring. If you add or multiply any two integers, the result is an integer. Also, to be a ring, the addition operator must satisfy associativity, commutativity, distributivity, and the multiplication operator must be associative as well. There must be an additive identity (a+0=a) and an additive inverse for every set element (-a+a=0). These are all basic concepts to which we are introduced in grade school. It's pretty neat to see that they are fundamental principles on which all of mathematics is built.